### Home > CALC > Chapter 9 > Lesson 9.2.2 > Problem9-67

9-67.

Thoroughly investigate the graph of f(x) = x5/3 + x2/3 . Identify all important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify point(s) of inflection, cusps, and intercepts. Be sure to justify all statements graphically and analytically. Homework Help ✎

$f^\prime(x)=\frac{5}{3}x^{2/3}+\frac{2}{3}x^{-1/3}=\frac{1}{3}x^{-1/3}(5x+2)$

Solve f′(x) = 0 to locate the extrema and determine the increasing/decreasing intervals.

$f^{\prime\prime}(x)=\frac{10}{9}x^{-1/3}+\frac{2}{9}x^{-4/3}=\frac{2}{9}x^{-4/3}(5x-1)$

Solve f''(x) = 0 to determine points of inflection and the intervals of concavity.